two function models g(t) and n(t) are being considered to represent the sales for a company manufacturing graphing calculators. both g(t) and n(t) represent the sales of the calculators, in thousands of units, after t years for $9 \le t \le 15$. if $g(t) = 11 - 2.144\ln(x)$ and $n(t) = -x + 15$, what is the first time t that sales predicted by the logarithmic model will be 0.5 thousand units more than sales predicted by the linear model?

9.275
9.881
10.432
12.116

Question

two function models g(t) and n(t) are being considered to represent the sales for a company manufacturing graphing calculators. both g(t) and n(t) represent the sales of the calculators, in thousands of units, after t years for \$9 \le t \le 15\$. if \$g(t) = 11 - 2.144\ln(x)\$ and \$n(t) = -x + 15\$, what is the first time t that sales predicted by the logarithmic model will be 0.5 thousand units more than sales predicted by the linear model?

9.275
9.881
10.432
12.116

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Logarithmic Models", "Transcendental Equations" ], "new_concepts": [ "Numerical Equation Solving" ], "current_concepts": [ "Logarithmic Models", "Transcendental Equations", "Numerical Equation Solving" ] } ## Set up the model equation Using the *Logarithmic Models* knowledge point $$ \begin{aligned} &G(t) = 11 - 2.144\ln(t)\ &N(t) = -t + 15\ &G(t) = N(t) + 0.5\ &11 - 2.144\ln(t) = -t + 15 + 0.5\ &t - 2.144\ln(t) - 4.5 = 0 \end{aligned} $$ ## Solve the transcendental equation Using the *Transcendental Equations* knowledge point $$ \begin{aligned} &f(t) = t - 2.144\ln(t) - 4.5 = 0\ &f(9.275) \approx 9.275 - 2.144(2.2273) - 4.5 \approx 9.275 - 4.775 - 4.5 = 0\ &f(9.881) \approx 9.881 - 2.144(2.2906) - 4.5 \approx 9.881 - 4.911 - 4.5 = 0.47\ &f(12.116) \approx 12.116 - 2.144(2.4945) - 4.5 \approx 12.116 - 5.348 - 4.5 = 2.268\ &t \approx 9.275 \end{aligned} $$

(A) 9.275

(B) 9.881

(C) 10.432

(D) 12.116

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Transcendental Equations" ] }

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Explanation:

Set up the model equation

Solve the transcendental equation

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